reserve A for preIfWhileAlgebra,
  C,I,J for Element of A;
reserve S for non empty set,
  T for Subset of S,
  s for Element of S;

theorem
  for A being associative with_catenation non-empty UAStr
  for I1,I2,I3 being Element of A holds (I1\;I2)\;I3 = I1\;(I2\;I3)
proof
  let A be associative with_catenation non-empty UAStr;
  let I1,I2,I3 be Element of A;
  reconsider f = (the charact of A).2 as
  2-ary associative non empty homogeneous
  quasi_total PartFunc of (the carrier of A)*, the carrier of A by Def14;
A1: 2 in dom the charact of A by Def11;
  arity f = 2 by COMPUT_1:def 21;
  then
A2: dom f = 2-tuples_on the carrier of A by COMPUT_1:22;
A3: In(2, dom the charact of A) = 2 by A1,SUBSET_1:def 8;
A4: <*I1,I2*> in dom f by A2,FINSEQ_2:137;
A5: <*I2,I3*> in dom f by A2,FINSEQ_2:137;
A6: <*I1,I2\;I3*> in dom f by A2,FINSEQ_2:137;
  <*I1\;I2,I3*> in dom f by A2,FINSEQ_2:137;
  hence thesis by A3,A4,A5,A6,Def2;
end;
